The formula for the ellipse, what is the formula for the ellipse

The standard equation for an ellipse.

The standard equation for an ellipse is divided into two cases:

When the focus is on the x-axis, the standard equation for an ellipse is:

When the focus is on the y-axis, the standard equation for an ellipse is:

The basic formula for an ellipse is: area s = (pi) a b, perimeter c = 2 b + 4 (a-b).

Case 1: If the focus is on the x-axis, the basic formula of the ellipse is x2 a+ y2 b=1 (a>b>0) (note: it is the square of x and the square of y), the focus coordinate f1(-c,0) f2(c,0) axis of symmetry takes the coordinate axis as the axis of symmetry, takes the origin as the center of symmetry, and the fixed-point coordinates a1(-a,0) a2(a,0) ,b1(0,b) b2(0,-b), the major axis 2a, the minor axis 2b, the range -a x a -b y b, and the eccentricity e=c a (0<>

Case 2: If the focus is on the y-axis, the basic formula for an ellipse is y2 a+ x2 b=1 (a>b>0), (Note: is the square of x and the square of y).

Focus coordinates f1(0, -c) f2(0, c), axis of symmetry with the coordinate axis as the axis of symmetry, with the origin as the center of symmetry, fixed-point coordinates a1(0, -a) a2(0, a) ,b2(b,0) b1(-b,0), major axis 2a, minor axis 2b range -a y a -b x b, eccentricity e=c a (0

The relationship between a, b, and c in the ellipse formula is a 2 = b 2 + c 2 (a > b > 0).

The major axis is 2a, the short axis is 2b, and the focal length is 2c.

The ellipse is the sum of the distances from the plane to the fixed points f1 and f2 equal to the constant (greater than |f1f2|The trajectories of the moving point p, f1 and f2 are called the two foci of the ellipse. The mathematical expression is: |pf1|+|pf2|=2a（2a>|f1f2|）。

The formula for ellipse circumference: l=2 b+4(a-b).

According to the first definition of an ellipse, a is used to denote the length of the major semi-axis of the ellipse, b is the length of the minor semi-axis of the ellipse, and a>b>0.

Ellipse circumference theorem: The circumference of an ellipse is equal to the circumference of the ellipse whose minor semi-axis length is the radius (2 b) plus four times the difference between the length of the major semi-axis of the ellipse (a) and the length of the minor semi-axis (b).

Ellipse Area Formula: s= ab

Ellipse area theorem: The area of an ellipse is equal to the product of pi ( ) multiplied by the length of the major semi-axis of the ellipse (a) and the length of the minor semi-axis (b).

Ellipse: x 2 a 2 + y 2 b 2 = 1 (a>b>0) with the focus on the x-axis.

or y2a2+x2b2=1(a>b>1) with the focus on the y-axis.

Hyperbola: x 2 a 2-y 2 b 2 = 1, the real axis is the x-axis, and the imaginary axis is the y-axis. y 2 b 2-x 2 a 2 = 1, the real axis is the y axis, and the imaginary axis is the x-axis.

Parabolic: Parabolic standard equation: y 2 = 2px

Parabola: y 2 = px or x 2 = py

s= (pi) a b (where a and b are the semi-major axis of the ellipse and the length of the semi-minor axis, respectively), or s= (circumference) a b 4 (where a and b are the long axis of the ellipse, respectively).

The formula for calculating the circumference of the ellipse is: l=t(r+r).

t is the elliptic coefficient, which can be found by the value of r r, and the value of the coefficient t can be found in the table; r is the short radius of the ellipse; r is the long radius of the ellipse.

Ellipse Circumference Theorem: The circumference of an ellipse is equal to the product of the sum of the short and long radii of the ellipse and the coefficient of the ellipse (including the perfect circle).

Proof that the circumference of an ellipse is equal to the length of a particular sinusoidal curve in a period:

A cylinder with radius r intersects with an inclined plane to obtain an ellipse, and the angle between the inclined plane and the horizontal plane is , and a circle with a short diameter over the ellipse is intercepted. Rotate an angle starting from the intersection of the circle and the ellipse. Then the height of the point on the ellipse corresponding to the point perpendicular to the circle can be obtained f(c)=r tan sin(c r).

r: cylindrical radius;

the angle of the plane on which the ellipse is located and the horizontal plane;

c: the corresponding arc length (moving from a certain intersection to a certain direction);

The above is a brief process of proof, then the circumference of the ellipse (x*cos) 2+y 2=r 2 is equal to the length of the sinusoidal curve of f(c)=r tan sin(c r) in one period, and the circumference of a circle t=2 r is exactly the circumference of a circle.

(x 2) a 2+(y 2) b 2=1 This is the standard equation.

In the case of parametric equations, x=acos@, y=bsin@